Algebraic Equation

An algebraic equation is an equation expressed in the form:

$$ P(x) = 0 $$

where ( P(x) ) is a polynomial.

For example:

$$ x + 8 = 0 $$

Here, ( x + 8 ) is a polynomial, and therefore the equation is called a polynomial equation.

An algebraic equation is always balanced, meaning that the values on both sides of the equation are equal. It generally consists of variables, coefficients, and constants.

Consider the equation:

$$ 1 + 1 = 2 $$

This is balanced, since both sides have the same value. When manipulating algebraic equations, any operation performed on one side must also be performed on the other side to preserve balance.

For example, adding 5 to both sides:

$$ 1 + 1 + 5 = 2 + 5 $$

The same principle applies for subtraction, multiplication, and division. As long as identical operations are performed on both sides, the equation remains balanced.

The solution of an algebraic equation involves finding values of the variables that satisfy the equation. Such values are called the roots of the equation.

Transcendental Equation

A transcendental equation is an equation that contains transcendental functions such as exponential, logarithmic, trigonometric, or inverse trigonometric functions.

Unlike algebraic equations, transcendental equations typically do not have closed-form solutions and often require numerical or iterative methods to solve.

Examples:

$$ x = e^{-x}, \quad x = \cos(x), \quad 2x = x^2 $$

Key Differences Between Algebraic and Transcendental Equations

Feature	Algebraic Equation	Transcendental Equation
Functions Used	Polynomials only	Exponential, logarithmic, trigonometric, etc.
Solutions	Often exact (closed-form)	Usually approximate (numerical methods)
Example	( x^2 - 5x + 6 = 0 )	( x = \cos x )

Insight

• Algebraic equations are simpler to solve since they deal with polynomials, and exact roots can often be found using formulas (e.g., quadratic formula).
• Transcendental equations are more complex and require iterative techniques for finding approximate solutions, as exact solutions are rare.
• In real-life applications:
  • Algebraic equations appear in problems like motion, finance, and geometry.
  • Transcendental equations arise in physics, engineering, signal processing, and control systems.